Regularity of the level set flow

TL;DR

This paper characterizes the regularity of the level set flow, showing that the second derivative's continuity depends on the flow having a single singular time with a specific singular set structure.

Contribution

It establishes a precise condition linking the continuity of the second derivative of the level set function to the flow's singularity structure.

Findings

Second derivative is continuous iff there is a single singular time.

Singular set is a closed $C^1$ manifold with cylindrical singularities.

Flow becomes extinct at a single singular time.

Abstract

We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed $C^1$ manifold with cylindrical singularities.